Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 9

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x \cdot x, 0.04481, \mathsf{fma}\left(0.99229, x, 1\right)\right)} - x\right) \cdot 0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (- (/ (fma x 0.27061 2.30753) (fma (* x x) 0.04481 (fma 0.99229 x 1.0))) x)
  0.70711))

C

double code(double x) {
	return ((fma(x, 0.27061, 2.30753) / fma((x * x), 0.04481, fma(0.99229, x, 1.0))) - x) * 0.70711;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(x, 0.27061, 2.30753) / fma(Float64(x * x), 0.04481, fma(0.99229, x, 1.0))) - x) * 0.70711)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * 0.04481 + N[(0.99229 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{x \cdot \color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x\right) \]

   5. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{x \cdot \color{blue}{\left(x \cdot \frac{4481}{100000} + \frac{99229}{100000}\right)} + 1} - x\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\left(x \cdot \frac{4481}{100000}\right) \cdot x + \frac{99229}{100000} \cdot x\right)} + 1} - x\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\left(x \cdot \frac{4481}{100000}\right) \cdot x + \color{blue}{x \cdot \frac{99229}{100000}}\right) + 1} - x\right) \]

   8. associate-+l+ N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot \frac{4481}{100000}\right) \cdot x + \left(x \cdot \frac{99229}{100000} + 1\right)}} - x\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(x \cdot \frac{4481}{100000}\right)} + \left(x \cdot \frac{99229}{100000} + 1\right)} - x\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{x \cdot \color{blue}{\left(x \cdot \frac{4481}{100000}\right)} + \left(x \cdot \frac{99229}{100000} + 1\right)} - x\right) \]

   11. associate-*r* N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot x\right) \cdot \frac{4481}{100000}} + \left(x \cdot \frac{99229}{100000} + 1\right)} - x\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{4481}{100000}, x \cdot \frac{99229}{100000} + 1\right)}} - x\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{4481}{100000}, x \cdot \frac{99229}{100000} + 1\right)} - x\right) \]

   14. *-commutative N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(x \cdot x, \frac{4481}{100000}, \color{blue}{\frac{99229}{100000} \cdot x} + 1\right)} - x\right) \]

   15. lower-fma.f64 99.9

       \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\mathsf{fma}\left(x \cdot x, 0.04481, \color{blue}{\mathsf{fma}\left(0.99229, x, 1\right)}\right)} - x\right) \]

4. Applied rewrites 99.9%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.04481, \mathsf{fma}\left(0.99229, x, 1\right)\right)}} - x\right) \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(x \cdot x, \frac{4481}{100000}, \mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)\right)} - x\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(x \cdot x, \frac{4481}{100000}, \mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)\right)} - x\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(x \cdot x, \frac{4481}{100000}, \mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)\right)} - x\right) \]

   4. lower-fma.f64 99.9

      \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(x \cdot x, 0.04481, \mathsf{fma}\left(0.99229, x, 1\right)\right)} - x\right) \]

6. Applied rewrites 99.9%

   \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(x \cdot x, 0.04481, \mathsf{fma}\left(0.99229, x, 1\right)\right)} - x\right) \]

7. Final simplification 99.9%

   \[\leadsto \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x \cdot x, 0.04481, \mathsf{fma}\left(0.99229, x, 1\right)\right)} - x\right) \cdot 0.70711 \]
8. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 -100.0)
     t_0
     (if (<= t_1 2.5)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= -100.0) {
		tmp = t_0;
	} else if (t_1 <= 2.5) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= -100.0)
		tmp = t_0;
	elseif (t_1 <= 2.5)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, -100.0], t$95$0, If[LessEqual[t$95$1, 2.5], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -100 or 2.5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if -100 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2.5

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      6. lower-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right)}, x, 1.6316775383\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -100:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -100.0)
     (* -0.70711 x)
     (if (<= t_0 2.5)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 2.5) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 2.5)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 2.5], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -100 or 2.5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 97.9

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -100 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2.5

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      6. lower-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right)}, x, 1.6316775383\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -100.0)
     (* -0.70711 x)
     (if (<= t_0 2.5)
       (fma x -2.134856267379707 1.6316775383)
       (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 2.5) {
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 2.5)
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 2.5], N[(x * -2.134856267379707 + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -100 or 2.5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 97.9

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -100 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2.5

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-2134856267379707}{1000000000000000}} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 2.5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -100.0)
     (* -0.70711 x)
     (if (<= t_0 2.5) 1.6316775383 (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 2.5) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.27061d0 * x) + 2.30753d0) / ((((0.04481d0 * x) + 0.99229d0) * x) + 1.0d0)) - x
    if (t_0 <= (-100.0d0)) then
        tmp = (-0.70711d0) * x
    else if (t_0 <= 2.5d0) then
        tmp = 1.6316775383d0
    else
        tmp = (-0.70711d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -100.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 2.5) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x
	tmp = 0
	if t_0 <= -100.0:
		tmp = -0.70711 * x
	elif t_0 <= 2.5:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 2.5)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = -0.70711 * x;
	elseif (t_0 <= 2.5)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 2.5], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -100 or 2.5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 97.9

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -100 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2.5

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{1.6316775383} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -100:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2.5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (- (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x)
  0.70711))

C

double code(double x) {
	return ((fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x) * 0.70711;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x) * 0.70711)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000}} \]

   3. lower-*.f64 99.9

      \[\leadsto \color{blue}{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \cdot 0.70711} \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   7. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   8. lower-fma.f64 99.9

      \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \cdot 0.70711 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x\right) \cdot \frac{70711}{100000} \]

   10. +-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x\right) \cdot \frac{70711}{100000} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x\right) \cdot \frac{70711}{100000} \]

   12. *-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x\right) \cdot \frac{70711}{100000} \]

   13. lower-fma.f64 99.9

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x\right) \cdot 0.70711 \]

   14. lift-+.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   15. +-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   16. lift-*.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   17. *-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   18. lower-fma.f64 99.9

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x\right) \cdot 0.70711 \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (/ (fma 0.27061 x 2.30753) (fma 0.99229 x 1.0)) x) 0.70711))

C

double code(double x) {
	return ((fma(0.27061, x, 2.30753) / fma(0.99229, x, 1.0)) - x) * 0.70711;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(0.27061, x, 2.30753) / fma(0.99229, x, 1.0)) - x) * 0.70711)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(0.99229 * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x\right) \]

   2. metadata-eval N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} \cdot 1\right)} \cdot x + 1} - x\right) \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) \cdot x + 1} - x\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)} \cdot x + 1} - x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\mathsf{fma}\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, x, 1\right)}} - x\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, 1\right)} - x\right) \]

   7. lft-mult-inverse N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\frac{99229}{100000} \cdot \color{blue}{1}, x, 1\right)} - x\right) \]

   8. metadata-eval 98.3

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\mathsf{fma}\left(\color{blue}{0.99229}, x, 1\right)} - x\right) \]

5. Applied rewrites 98.3%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.99229, x, 1\right)}} - x\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000}} \]

   3. lower-*.f64 98.3

      \[\leadsto \color{blue}{\left(\frac{2.30753 + x \cdot 0.27061}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711} \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   7. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   8. lower-fma.f64 98.3

      \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711 \]

7. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711} \]
8. Add Preprocessing

Alternative 8: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2.30753}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (/ 2.30753 (fma 0.99229 x 1.0)) x) 0.70711))

C

double code(double x) {
	return ((2.30753 / fma(0.99229, x, 1.0)) - x) * 0.70711;
}

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 / fma(0.99229, x, 1.0)) - x) * 0.70711)
end

Wolfram

code[x_] := N[(N[(N[(2.30753 / N[(0.99229 * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x\right) \]

   2. metadata-eval N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} \cdot 1\right)} \cdot x + 1} - x\right) \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right) \cdot x + 1} - x\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)} \cdot x + 1} - x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\mathsf{fma}\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, x, 1\right)}} - x\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, 1\right)} - x\right) \]

   7. lft-mult-inverse N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\frac{99229}{100000} \cdot \color{blue}{1}, x, 1\right)} - x\right) \]

   8. metadata-eval 98.3

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\mathsf{fma}\left(\color{blue}{0.99229}, x, 1\right)} - x\right) \]

5. Applied rewrites 98.3%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.99229, x, 1\right)}} - x\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{100000}}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{2.30753}}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \]

9. Final simplification 98.3%

   \[\leadsto \left(\frac{2.30753}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711 \]
10. Add Preprocessing

Alternative 9: 50.4% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
4. Step-by-step derivation

5. Applied rewrites 48.0%

   \[\leadsto \color{blue}{1.6316775383} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))